↑

Properties

Label 2-33488-1.1-c1-0-8
Degree $2$
Conductor $33488$
Sign $1$
Analytic cond. $267.403$
Root an. cond. $16.3524$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 3·9-s − 13-s + 6·17-s + 4·19-s + 23-s − 5·25-s − 2·29-s + 10·31-s + 4·37-s + 10·41-s + 4·43-s − 6·47-s + 49-s − 2·53-s − 2·59-s + 10·61-s − 3·63-s + 16·67-s − 10·71-s − 2·73-s + 8·79-s + 9·81-s − 16·83-s − 8·89-s − 91-s − 4·97-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.208·23-s − 25-s − 0.371·29-s + 1.79·31-s + 0.657·37-s + 1.56·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s − 0.274·53-s − 0.260·59-s + 1.28·61-s − 0.377·63-s + 1.95·67-s − 1.18·71-s − 0.234·73-s + 0.900·79-s + 81-s − 1.75·83-s − 0.847·89-s − 0.104·91-s − 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33488\)    =    \(2^{4} \cdot 7 \cdot 13 \cdot 23\)
Sign: $1$
Analytic conductor: \(267.403\)
Root analytic conductor: \(16.3524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33488,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.425554745\)
\(L(\frac12)\) \(\approx\) \(2.425554745\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.80321150285104, −14.40418173938190, −14.12373053605075, −13.53151283779567, −12.88578564005266, −12.24183256670287, −11.79479985516966, −11.36892568189107, −10.90352114818684, −9.985699698773880, −9.760427117100390, −9.171434981777035, −8.317554577150071, −8.003390250439646, −7.529393021603460, −6.801233130377701, −5.934980023170577, −5.674675432902057, −5.040281392154750, −4.321549024373820, −3.580774897031827, −2.895080173998388, −2.391868966084426, −1.319854618457606, −0.6334062801930164, 0.6334062801930164, 1.319854618457606, 2.391868966084426, 2.895080173998388, 3.580774897031827, 4.321549024373820, 5.040281392154750, 5.674675432902057, 5.934980023170577, 6.801233130377701, 7.529393021603460, 8.003390250439646, 8.317554577150071, 9.171434981777035, 9.760427117100390, 9.985699698773880, 10.90352114818684, 11.36892568189107, 11.79479985516966, 12.24183256670287, 12.88578564005266, 13.53151283779567, 14.12373053605075, 14.40418173938190, 14.80321150285104

Graph of the $Z$-function along the critical line